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Towngate Primary Academy Calculation Policy Our Mathematics curriculum at Towngate Primary Academy The curriculum must raise standards. We have to avoid shallow mastery and ensure that we help children to embed concepts so they can apply their understanding of maths. Children need to have the language of learning and by using the correct vocabulary this shows confidence and depth of understanding. A definition of a Mathematician: Someone who can see patterns. Someone who shows deeper application. Someone who identifies and understands the interconnectivity of concepts and demonstrates this through the transfer of skills. Someone who is systematic and resilient and can extend their own learning. EXPECTATIONS FOR PROGRESS Children should be allowed to progress through the calculation process for each operation as and when they are ready to move on. However, the next few sheets set out the expected progress in each year group. ADDITION Reception Year 1 Counting in ones, 1:1 correspondence starting from difference numbers – up to 20. Adding 2 groups together by counting and counting on. Count to, read and write numbers across 100. Use partitioning and add 2 digit number. Number bonds 10, 20 & 100. Apply written methods as well as concrete objects. Add 1 & 2 digit numbers to 20 including 0. Adding 3 digit numbers using partitioning. Solve one step problems that involve addition using concrete objects and mentally. Doubling & halving simple numbers. Understanding of commutative law in relation to addition. Missing & number problems. Use language of equal to, more than. Given a number identify 1 more or 1 less. Partition 2 digit numbers into tens and units. Add simple 2 digit numbers together Year 2 Use inverse to check missing number problems. Doubling & halving including multiples of 12. Extend mental maths strategies to include number bonds. Year 3 Year 4 Year 5 Year 6 Partition using columns for addition – involve crossing 10 then 100. Extend to 4 digit numbers when secure. Formal column method of addition (4 digit numbers). Introduce adding decimal in a column. Add negative integers. Involve 2 step problems. Consolidating & applying knowledge to solve problems. Extend children who are ready to formal column method Counting in multiples of 4, 8, 50 & 100. (6, 7, 9, 25 & 1000 extension) Adding 3 lots of four digit numbers. Read, write & compare numbers to at least 1,000,000. Interpret negative numbers in context, calculate intervals across zero. Solve number problems & practical problems. Compare & order number to 1000 (and beyond). Solve number and practical problems involving these ideas. Doubling & halving 2, 3 & 4 digit number (odd numbers). SUBTRACTION Reception Year 1 Physically taking Subtract by away using finding the numbers up to 20. difference on a number line. Using number Numbers should lines with physical extend as objects. children become more confident. This then needs applying to problems both written and practical. Missing number sentences. Application to number challenges using inverse to check. Year 2 Year 3 Subtract by finding the difference on a number line. Begin to do larger jumps of 10 or 2. Subtract by finding the difference on a number line. Use a number line to make bigger jumps. Mixture of numbers counting onto the next whole 10, 100. Extension work to involve 3 digit numbers. Extend children who are ready to formal column method. Doubling / halving 2, 3 and 4 digit number. Application to number challenges using inverse to check. Application to number challenges using inverse to check. Year 4 Year 5 Year 6 Subtract using formal column method. Subtract using formal column method. Subtract using formal column method. Application to number challenges using inverse to check. Decimals (as money) Decimals (as money) Application to number challenges using inverse to check. Application to number challenges using inverse to check. MULTIPLICATION Year 1 Year 2 Solve simple one step problems involving ‘group of’ concrete and pictorial objects. 2, 5, 10 times table and understand it as repeated addition. Year 3 Children should know all times tables by end of year. Learn these tables, extend Multiplication using grid to 3, 4 when confident. method, 2 digit x 1 digit. Extend to larger numbers to challenge higher ability. Solve problems using Application to number materials, array & challenges. Real life repeated addition. situations & problems. Calculate simple number sentences using table they know – begin to use grid method with higher ability. Understand cumulative law with x link to +. Application to number challenges. Real life situations & problems. Year 4 Year 5 & Year 6 Consolidate all times tables. Consolidate all times tables. Introduce multiplication in formal method. Multiply multi digit numbers up to 4 digit whole numbers using formal method. Multiply decimal numbers by 10, 100 and 1000. 2 & 3 digit x 1 & digit. Extend to 4 addition in columns. Application to number challenges. Real life situations & problems. Identify multiples, factors, common factors and prime numbers. Recognise squared and cubed numbers. Application to number challenges. Real life situations & problems. DIVISION Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Solve simple one step problems involving division using concrete / pictorial objects. Using sharing to understand the concept. Application into number challenges, use invers of known x tables to check answers. Share between physically into groups, then put onto a number line. Divide using formal method starting at 0. Use place value to recall multiplication and division facts for all tables. Use place value to recall multiplication and division facts for all tables. Use place value to recall multiplication and division facts for all tables. Larger numbers. Calculate with small remainders when confident. Larger number. Divide mentally using known facts. Divide mentally using known facts. Divide mentally using known facts. Use times tables to divide by 2 & 3 digit number. Use times tables to divide 4 digit by 2 & 3 digit numbers. Use times tables to divide 4 digit by 2 & 3 digit numbers. Give remainders as a fraction / decimal. Use knowledge of BODMAS to carry out calculations. Give remainders as a fraction / decimal. Simple remainders. To understand the inverse to prove it. Apply to fractions. Application into number challenges, use invers of known times tables to check answers. Application into number challenges, use invers of times tables to check answers. Foundation Stage Mathematics Practitioners in Foundation Stage follow the mathematics strands from the Early Years Foundation Stage Framework. Children are supported in developing their understanding of mathematics (number calculation, shape, space and measure) in a broad range of contexts in which they can explore, enjoy, learn, practice and talk about their developing understanding. Children are actively engaged in their learning in order to process it meaningfully. This is promoted through focus activities and the children’s individual interests. By the end of foundation stage children are expected to: Count reliably with numbers from one to 20, place them in order and say which number is one more or one less than a given number. Using quantities and objects, they add and subtract two single-digit numbers and count on or back to find the answer. They solve problems, including doubling, halving and sharing. Use everyday language to talk about size, weight, capacity, position, distance, time and money to compare quantities and objects and to solve problems. They recognize, create and describe patterns. They explore characteristics of everyday objects and shapes and use mathematical language to describe them. The classroom environment provides real life practical opportunities to support the application of mathematics. Focus teaching is based around real life scenarios, rhymes, songs and books. Practical apparatus such as real life objects, numicon and links are used to support this. Year 1 Age Related Expectations Addition Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line Key skills for addition at Year 1: Read and write numbers to 100 in numerals, incl. 1—20 in words Recall bonds to 10 and 20, and addition facts within 20 Count to and across 100 Count in multiples of 1 2, 5 and 10 Solve simple 1-step problems involving addition, using objects, number lines and pictorial representations. Subtraction Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is_? Key skills for subtraction at Year 1: Given a number, say one more or one less. Count to and over 100, forward and back, from any number. Represent and use subtraction facts to 20 and within 20. Subtract with one-digit and two-digit numbers to 20, including zero. Solve one-step problems that involve addition and subtraction, using concrete objects (ie bead string, objects, cubes) and pictures, and missing number problems. Read and write numbers from 0 to 20 in numerals and words. Multiplication Key vocabulary: groups of, lots of, times, array, altogether, multiply, count Key skills for multiplication at Year1: Count in multiples of 2, 5 and 10. Solve one-step problems involving multiplication, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher. Make connections between arrays, number patterns, and counting in twos, fives and tens. Begin to understand doubling using concrete objects and pictorial representations. Division Key Vocabulary: share, share equally, one each, two each…, group, groups of, lots of, array Key number skills needed for division at Year 1: Solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations arrays with the support of the teacher Through grouping and sharing small quantities, pupils begin to understand, division, and finding simple fractions of objects, numbers and quantities. They make connections between arrays, number patterns, and counting in twos, fives and tens. Year 2 Age Related Expectations Addition Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, addition, column, tens boundary Key skills for addition at Year 2: Add a 2-digit number and ones (e.g. 27 + 6) Add a 2-digit number and tens (e.g. 23 + 40) Add pairs of 2-digit numbers (e.g. 35 + 47) Add three single-digit numbers (e.g. 5 + 9 + 7) Show that adding can be done in any order (the commutative law). Recall bonds to 20 and bonds of tens to 100 (30 + 70 etc.) Count in steps of 2, 3 and 5 and count in tens from any number. Understand the place value of 2-digit numbers (tens and ones) Compare and order numbers to 100 using < > and = signs. Read and write numbers to at least 100 in numerals and words. Solve problems with addition, using concrete objects, pictorial representations, involving numbers, quantities and measures, and applying mental and written methods. Subtraction Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back , how many left, how much less is_? difference, count on, strategy, partition, tens, units Key skills for subtraction at Year 2: Recognise the place value of each digit in a two-digit number. Recall and use subtraction facts to 20 fluently, and derive and use related facts up to 100. Subtract using concrete objects, pictorial representations, 100 squares and mentally, including: a two-digit number and ones, a two-digit number and tens, and two two-digit numbers. Show that subtraction of one number from another cannot be done in any order. Recognise and use inverse relationship between addition and subtraction, using this to check calculations and missing number problems. Solve simple addition and subtraction problems including measures, using concrete objects, pictorial representation, and also applying their increasing knowledge of mental and written methods. Read and write numbers to at least 100 in numerals and in words. Multiplication Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, times as big as, once, twice, three times... Key skills for multiplication at Year 2: Count in steps of 2, 3 and 5 from zero, and in 10s from any number. Recall and use multiplication facts from the 2, 5 and 10 multiplication tables, including recognising odds and evens. Write and calculate number statements using the x and = signs. Show that multiplication can be done in any order (commutative). Solve a range of problems involving multiplication, using concrete objects, arrays, repeated addition, mental methods, and multiplication facts. Pupils use a variety of language to discuss and describe multiplication. Division Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over Key number skills needed for division at Year 2: Count in steps of 2, 3, and 5 from 0 Recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers. Calculate mathematical statements for multiplication and division within the multiplication tables and write them using the x, ÷ and = signs. Show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot. Solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts. Year 3 Age Related Expectations Addition Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens boundary, hundreds boundary, increase, vertical, ‘carry‘, expanded, compact Key skills for addition at Year 3: Read and write numbers to 1000 in numerals and words. Add 2-digit numbers mentally, incl. those exceeding 100. Add a three-digit number and ones mentally (175 + 8) Add a three-digit number and tens mentally (249 + 50) Add a three-digit number and hundreds mentally (381 + 400) Estimate answers to calculations, using inverse to check answers. Solve problems, including missing number problems, using number facts, place value, and more complex addition. Recognise place value of each digit in 3-digit numbers (hundreds, tens, ones.) Continue to practise a wide range of mental addition strategies, ie. number bonds, adding the nearest multiple of 10, 100, 1000 and adjusting, using near doubles, partitioning and recombining. Subtraction Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is_? difference, count on, strategy, partition, tens, units exchange, decrease, hundreds, value, digit Key skills for subtraction at Year 3: Subtract mentally a: 3-digit number and ones, 3-digit number and tens, 3-digit number and hundreds. Estimate answers and use inverse operations to check. Solve problems, including missing number problems. Find 10 or 100 more or less than a given number. Recognise the place value of each digit in a 3-digit number. Counting up differences as a mental strategy when numbers are close together or near multiples of 10 (see examples above) Read and write numbers up to 1000 in numerals and words. Practise mental subtraction strategies, such as subtracting near multiples of 10 and adjusting (e.g. subtracting 19 or 21), and select most appropriate methods to subtract, explaining why. Multiplication Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, times, _times as big as, once, twice, three times..., partition, grid method, multiple, product, tens, units, value Key skills for multiplication at Year 3: Recall and use multiplication facts for the 2, 3, 4, 5, 8 and 10 multiplication tables, and multiply multiples of 10. Write and calculate number statements using the multiplication tables they know, including 2digit x single digit, drawing upon mental methods, and progressing to reliable written methods. Solve multiplication problems, including missing number problems. Develop mental strategies using commutativity (e.g. 4 x 12 x 5 = 4 x 5 x 12 = 20 x 12 = 240) Solve simple problems in contexts, deciding which operations and methods to use. Develop efficient mental methods to solve a range of problems e.g using commutativity (4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240) and for missing number problems ? x 5 = 20, 3 x ? = 18, ? x ? = 32 Division Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, ‘carry‘, remainder, multiple Key number skills needed for division at Year 3: Recall and use multiplication and division facts for the 2, 3, 4, 5, 8 and 10 multiplication tables (through doubling, connect the 2, 4 and 8s). Write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to for-mal written methods. Solve problems, in contexts, and including missing number problems, involving multiplication and division. Pupils develop efficient mental methods, for example, using multiplication and division facts (e.g. using 3 × 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3) to derive related facts (30 × 2 = 60, so 60 ÷ 3 = 20 and 20 = 60 ÷ 3). Pupils develop reliable written methods for division, starting with calculations of 2-digit numbers by 1-digit numbers and progressing to the formal written method of short division. Year 4 Age Related Expectations Addition Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens boundary, hundreds boundary, increase, vertical, „carry‟, expanded, compact, thousands, hundreds, digits, inverse Key skills for addition at Year 4: Select most appropriate method: mental, jottings or written and explain why. Recognise the place value of each digit in a four-digit number. Round any number to the nearest 10, 100 or 1000. Estimate and use inverse operations to check answers. Solve 2-step problems in context, deciding which operations and methods to use and why. Find 1000 more or less than a given number. Continue to practise a wide range of mental addition strategies, i.e. number bonds, add the nearest multiple of 10, 100, 1000 and adjust, use near doubles, partitioning and recombining. Add numbers with up to 4 digits using the formal written method of column addition Solve 2-step problems in contexts, deciding which operations and methods to use and why. Estimate and use inverse operations to check answers to a calculation. Subtraction Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance be-tween, how many more, how many fewer / less than, most, least, count back, how many left, how much less is_? difference, count on, strategy, partition, tens, units exchange, decrease, hundreds, value, digit, inverse Key skills for subtraction at Year 4: Subtract by counting on where numbers are close together or they are near to multiples of 10, 100 etc. Children select the most appropriate and efficient methods for given subtraction calculations. Estimate and use inverse operations to check answers. Solve addition and subtraction 2-step problems, choosing which operations and methods to use and why. Solve simple measure and money problems involving fractions and decimals to two decimal places. Find 1000 more or less than a given number. Count backwards through zero, including negative numbers. Recognise place value of each digit in a 4-digit number Round any number to the nearest 10, 100 or 1000 Solve number and practical problems that involve the above, with increasingly large positive numbers. Multiplication Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, array, column, row, commutative, groups of, sets of, lots of, equal groups, times, multiply, times as big as, once, twice, three times... partition, grid method, total, multiple, product, sets of, inverse Key skills for multiplication at Year 4: Count in multiples of 6, 7, 9, 25 and 1000 Recall multiplication facts for all multiplication tables up to 12 x 12. Recognise place value of digits in up to 4-digit numbers Use place value, known facts and derived facts to multiply mentally, e.g. multiply by 1, 10, 100, by 0, or to multiply 3 numbers. Use commutativity and other strategies mentally 3 x 6 = 6 x 3 , 2 x 6 x 5 = 10 x 6 , 39x7 = 30 x 7 + 9 x 7. Solve problems with increasingly complex multiplication in a range of contexts. Count in multiples of 6, 7, 9, 25 and 1000 Recognise the place value of each digit in a four-digit number (thousands, hundreds, tens, and units) Division Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, “carry‟, remainder, multiple, divisible by, factor Key number skills needed for division at Year 4: Recall multiplication and division facts for all numbers up to 12 x 12. Use place value, known and derived facts to multiply and divide mentally, including: multiplying and dividing by 10 and 100 and 1. Pupils practise to become fluent in the formal written method of short division with exact answers when dividing by a one-digit number Pupils practise mental methods and extend this to three-digit numbers to derive facts, for example 200 × 3 = 600 so 600 ÷ 3 = 200 Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as three cakes shared equally between 10 children. Year 5 Age Related Expectations Addition Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens boundary, hundreds boundary, increase, „carry‟, expanded, compact, vertical, thousands, hundreds, digits, inverse & decimal places, decimal point, tenths, hundredths, thousandths Key skills for addition at Year 5: Add numbers mentally with increasingly large numbers, using and practising a range of mental strategies i.e. add the nearest multiple of 10, 100, 1000 and adjust; use near doubles, inverse, partitioning and re-combining; using number bonds. Use rounding to check answers and accuracy. Solve multi-step problems in contexts, deciding which operations and methods to use and why. Read, write, order and compare numbers to at least 1 million and determine the value of each digit. Round any number up to 1, 000, 000 to the nearest 10, 100, 1000, 10,000 and 100,000. Add numbers with more than 4 digits using formal written method of columnar addition. Subtraction Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is_? difference, count on, strategy, partition, tens, units exchange, decrease, hundreds, value, digit, inverse, tenths, hundredths, decimal point, decimal Key skills for subtraction at Year 5: Subtract numbers mentally with increasingly large numbers. Use rounding and estimation to check answers to calculations and determine, in a range of contexts, levels of accuracy. Solve addition and subtraction multi-step problems in context, deciding which operations and methods to use and why. Read, write, order and compare numbers to at least 1 million and determine the value of each digit. Count forwards or backwards in steps of powers of 10 for any given number up to 1 million. Interpret negative numbers in context, counting forwards and backwards with positive and negative integers through zero. Round any number up to 1 million to the nearest 10, 100, 1000, 10,000 and 100,000. Multiplication Key vocabulary groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, _times as big as, once, twice, three times..., partition, grid method, total, multiple, product, inverse, square, factor, integer, decimal, short/long multi-plication, ‘carry‘ Key skills for multiplication at Year 5: Identify multiples and factors, using knowledge of multiplication tables to 12x12. Solve problems where larger numbers are decomposed into their factors Multiply and divide integers and decimals by 10, 100 and 1000 Recognise and use square and cube numbers and their notation Solve problems involving combinations of operations, choosing and using calculations and methods appropriately. Division Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, “carry‟, remainder, multiple, divisible by, factor, inverse, quotient, prime number, prime factors, composite number (non-prime) Key number skills needed for division at Year 5: Recall multiplication and division facts for all numbers up to 12 x 12 (as in Y4). Multiply and divide numbers mentally, drawing upon known facts. Identify multiples and factors, including finding all factor pairs of a number, and common factors of two number. Solve problems involving multiplication and division where larger numbers are decomposed into their factors. Multiply and divide whole numbers and those involving decimals by 10, 100 and 1000. Use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers. Work out whether a number up to 100 is prime, and recall prime numbers to 19. Divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context Use multiplication and division as inverses. Interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding (e.g. 98 ÷ 4 = 24 r2 = 241/2 = 24.5 ≈ 25). Solve problems involving combinations of all four operations, including understanding of the equals sign, and including division for scaling by different fractions and problems involving simple rates. Year 6 Age Related Expectations Addition Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens boundary, hundreds boundary, increase, “carry‟, expanded, compact, vertical, thousands, hundreds, digits, inverse, decimal places, decimal point, tenths, hundredths, thousandths Key skills for addition at Year 6: Perform mental calculations, including with mixed operations and large numbers, using and practising a range of mental strategies. Solve multi-step problems in context, deciding which operations and methods to use and why. Use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy. Read, write, order and compare numbers up to 10 million and determine the value of each digit. Round any whole number to a required degree of accuracy. Pupils understand how to add mentally with larger numbers and calculations of increasing complexity. Subtraction Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back , how many left, how much less is_? difference, count on, strategy, partition, tens, units exchange, decrease, hundreds, value, digit, inverse, tenths, hundredths, decimal point, decimal Key skills for subtraction at Year 6: Solve addition and subtraction multi-step problems in context, deciding which operations and methods to use and why. Read, write, order and compare numbers up to 10 million and determine the value of each digit Round any whole number to a required degree of accuracy Use negative numbers in context, and calculate intervals across zero. Children need to utilise and consider a range of mental subtraction strategies, jottings and written methods before choosing how to calculate. Multiplication Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, array, column, row, commutative, sets of, equal groups, times as big as, once, twice, three times... partition, grid method, total, multiple, product, inverse, square, factor, integer, decimal, short / long multiplication, „carry‟, tenths, hundredths, decimal Key skills for multiplication at Y6: Recall multiplication facts for all times tables up to 12 x 12 (as Y4 and Y5). Multiply multi-digit numbers, up to 4-digit x 2-digit using long multiplication. Perform mental calculations with mixed operations and large numbers. Solve multi-step problems in a range of contexts, choosing appropriate combinations of operations and methods. Estimate answers using round and approximation and determine levels of accuracy. Round any integer to a required degree of accuracy. Division Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, “carry‟, remainder, multiple, divisible by, factor, inverse, quotient, prime number, prime factors, composite number (non-prime), common factor Key number skills needed for division at Year 6: Recall and use multiplication and division facts for all numbers to 12 x 12 for more complex calculations Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context. Use short division where appropriate. Perform mental calculations, including with mixed operations and large numbers. Identify common factors, common multiples and prime numbers. Solve problems involving all 4 operations. Use estimation to check answers to calculations and determine accuracy, in the context of a problem. Use written division methods in cases where the answer has up to two decimal places. Solve problems which require answers to be rounded to specified degrees of accuracy. Contents Introduction ....................................................................................... 3 Addition of 1 and 2-digit numbers ..................................................... 4 Addition of two 2-digit numbers ........................................................ 7 Addition of larger whole numbers ..................................................... 9 Addition of decimal numbers…………………………………………...9 Subtraction by taking away ............................................................. 10 Subtraction by finding the difference .............................................. 12 Multiplication ................................................................................... 14 Division ........................................................................................... 18 The Use of Calculators ................................................................... 22 Introduction This calculation policy sets out the methods that children will be taught and encouraged to use when tackling calculations as part of their daily numeracy lessons and during their work in a wide range of cross-curricular and real-life contexts. The policy reflects our belief that the methods taught should make sense to children and should be both efficient and reliable. ADDITION Addition of 1 and 2-digit numbers Children are introduced to addition as the adding of objects to another set1 of objects. For example, for the calculation five add two, like this. Children physically add objects to a set pictorially: the new set would look Children record like this: objects added The next step is for children to complete a 'number sentence2' by writing the correct numerals in the boxes. For example, for the sum five add two, the children would be given: add 5 add altogether is the same as 2 is the same as 7 altogether 1 a set is a group of objects which can be described using a number eg 3 beads is a set 2 a number sentence is a set of numbers and symbols that make mathematical sense eg 15 + 3 = 18 or 3 x 4 =12 The words are then replaced by symbols: 5 + = 2 7+ The children will be taught that the digit 3 '5' stands for a number of objects. It is important to stress that the '=' sign means 'is the same number as' So: 5 + 2 is the same number as 7 Children are taught that the same total is made whichever order the numbers are added in Children are taught that the same total is made whichever order the numbers are added in any order: the commutative4 property of addition: So 5+2=7 and 2+5=7 Children are taught to read number sentences where the numbers and symbols are positioned as follows: 5 + 2 = 7 7 = 5 + 2 2 + 5 = 7 7 = 2 + 5 Children are taught that to add one number to another mentally they should hold the largest number in their heads and count on the smallest number. The children will say: '5 ...... 6, 7' At the same time children use a number line5 to perform addition calculations. For example, for the sum 'five add two' children use the number line like this: 0 1 2 3 4 5 6 7 8 9 10 They start at 5, make two jumps forward (add 2) then finish at 7. They record this as before: 5 + 2 = 7 At first children use number lines with numbers from 0 to 10 then move on to lines with numbers from 0 to 20. 3 a digit is any number symbol from 0 to 9 4 The commutative property of addition means that the numbers can be added in any order to give the same total 5 a number line represents a set of consecutive numbers eg 0 to 10, 100 to 200, -30 to 0 etc When adding using numbers from 20 to 100 children use relevant sections of a number line: eg: 24 + 8 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 They record the calculation as: 24 + 8 = 32 Addition bonds to 10 and 20 Children should know addition pairs that make 10 so that they can use their knowledge in their calculations. Children are taught the pattern made by the numbers: 10 + 0 = 10 9 + 1 = 10 8 + 2 = 10 7 + 3 = 10 6 + 4 = 10 5 + 5 = 10 4 + 6 = 10 3 + 7 = 10 2 + 8 = 10 1 + 9 = 10 0 +10 = 10 Children are taught to use the above addition pairs to quickly add three numbers where two of the numbers total 10. For example: 3 + 7 + 5 = 10 When children know that 3 + 7 = 10 they only need to add the remaining number - 5: 3 + 7 + 5 = 15 The same pattern is seen for addition pairs that total 20. Addition of two 2-digit numbers In preparation for the addition of two- and three- digit numbers children are taught to recognise the pattern in our number system: For example: 5 50 500 5000 + 2= 7 + 20 = 70 + 200 = 700 + 2000 = 7000 When learning how to add two 2-digit numbers6 children are taught how to break up or partition7 numbers. This is shown visually For example 46 is partitioned as 4 tens and 6 ones - 40 and 6 Children write this as 46 / \ 10 1 10 1 10 1 10 1 1 1 6 a 2-digit number is any number between 10 and 99 7 see note on page 6 25 is partitioned as 20 and 5 Children write this as 25 10 1 10 1 1 1 1 Children are taught to add the tens first For example 46 + 25 = 40 6 + 20 5 _________ 60 11 So 60 + 11 = 71 Children are taught to use the 100 number square8 to add numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Children move down one row for every 10 and one square to the right for every unit added. For example 46 + 25 Start on 46, add 20 by moving down two rows, then add 5 by moving five numbers to the right. Addition of larger whole numbers Formal method of addition: 368 + 493 For example + 368 493 1 1 __ 861 Starting at Units: 8+3 = 11, 1 unit and 1 ten to put into 10’s column. Tens: 60+90+10=160, 6 tens and 1 hundred to put into 100’s column. Hundreds: 300+400+100=800 Talking about the value of the numbers ensures that the children do not think they are ‘carrying one’. 8 the 100 number square shows the number line 1 to 100 in rows of 10 Addition of decimal numbers Children need to have a secure understanding of the place value of the numbers involved: 46 • 53 is made up of: For example 4 tens, 6 units, 5 tenths, 3 hundredths Addition of numbers that have one decimal place Children are introduced to addition of numbers that have one decimal place in the context of money: For example: 46 • 7 + 37 • 8 In the context of money: 46 and 37 are £s In the context of money: •7 and •8 are seen as 70p and 80p + 46.70 37.80 1 1 84.50 Always start with the lowest value: hundredths or pennies to explain the concept. This is the discussion around the calculating:0+0 hundredths =0 70+10 tenths = 150 or 1.50 so 50 to stay in tenths column and £1.00 or 1 whole unit to units column. 6+7+1 = 14 4 units and 1 ten. 40+3+10 = 80 If the number of pence is greater than 100, children need to understand that there is an additional pound. Addition of numbers that have two decimal places Children are introduced to addition of numbers that have two decimal places also in the context of money: £46 • 53 + £31 • 28 For example: + 46.53 31.28 1 77.81 Always start with the lowest value: hundredths or pennies to explain the concept. This is the discussion around the calculating:3+8 hundredths/pennies 50+20+10 = 11 = 80 6+1 40+30 =7 = 70 1 penny or hundredth and 1 ten pence. No need to move columns as it is below a whole one. units or pounds. pounds or tens. If the number of pence is greater than 100, children need to understand that there is an additional pound. NB In calculations where one of the numbers ‘shows’ more numbers after the decimal point than the other(s), children should use ‘0’ as the ‘place holder’ For example: If the calculation given is 57.2 + 29.43, children should understand that the first number can be recorded as 57.20 (the ‘0’ representing no hundredths). SUBTRACTION Subtraction by taking away Children are introduced to subtraction as removing ('taking away') objects from a set. They are taught to cross out the number of objects to be 'taken away' from a set and write the number of objects left. For example: 5 flowers take away 2 flowers leaves 3 flowers The words are then replaced by numerals to record subtraction calculations: For example: 5 - 2 = 3 Children are introduced to the mental strategy of counting backwards to perform subtraction calculations (the inverse of addition). For example: for the sum 'five take away two' children hold five 'in their head' and count backwards two numbers saying: 5 ........... 4, 3 The children are introduced to a number line to perform subtraction calculations. This helps them to understand that subtraction is the opposite (inverse) of addition. For example: 'five take away two' 0 1 2 3 4 5 6 7 8 9 10 They start at 5, make two jumps backwards (take away two) and finish on 3. They record: 5-2=3 Subtraction by finding the difference Children are taught the relationship between addition and subtraction, with the associated vocabulary. 4+2=6 4 add/plus 2 is the same number as/equals 6 2+4=6 2 add/plus 4 is the same number as/equals 6 6–2=4 6 take away/minus/subtract 2 equals 4 6–4=2 6 take away/minus/subtract 4 equals 2 Children are introduced to the concept of difference, and associated vocabulary. Finding the difference is the comparison of one number or set of objects with another. There are 2 more blue squares than orange squares. There are 2 less orange squares than blue squares. The difference between the number of blue squares and orange squares is 2. Children are then made aware of the link between subtraction and finding the difference. For example: 23 - 16 = The difference between 23 and 16 is 7 The difference between 16 and 23 is also 7 23 – 16 = 7 This is shown on a number line 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Provided the context 9 is appropriate, finding the difference between (rather than taking away) two numbers is the way children are encouraged to perform subtraction. 9 the context is the real-life background against which the calculation takes place eg shopping, baking etc The children count forward from the smaller number: 16 to the larger number: 23 as this is faster and more reliable than counting back from 23 to 16. Children are taught to use blank number lines For example: 364 - 241 9 241 50 250 60 300 4 360 364 Children add the 'jumps' to find the answer: 50 9 60 4 110 13 = 123 So 364 – 241 = 123 Formal column method of subtraction 365 – 246 = For example: 5 1 - 365 246 119 Always start with the smallest units: units 5 subtract 6 Without going into negative numbers I have not got enough so I need to take a ten from the 10’s column. This makes it 350+15 which still equals 365. Children need to understand that the number is not being changed. Now 15-6=9 Move onto tens column: 50-40=10 So that is 1 ten in the tens column. Move onto hundreds column: 300-200=100 So that is 1 hundred in the hundreds column. Subtraction involving decimal numbers Children are introduced to subtraction of numbers that have one decimal place in the context of money: For example: 96 • 4 - 24 • 6 In the context of money: 96 and 24 are £s In the context of money: •4 and •6 are seen as 40p and 60p The calculation becomes: £96.40 - £24.60 5 - 1 96.40 24.60 71.80 Always start with the smallest units: hundredths or pennies in this case 0-0= 0 Move onto tenths or 10 pences: 40-60= Without going into negative numbers I have not got enough so I need to take 1 unit or 1 pound to help me do this. This becomes 95.00 + 1.40 = 96.40. The number is not being changed. Now 1.40-0.60 (one pound forty subtract 60 pence) = 0.80 Move onto units or pounds: 5-4= 1 Move onto tens or ten pounds: 90-20= 70 Once children understand subtraction in the context of money they need to apply their knowledge and understanding to perform any subtraction calculation involving numbers with 2 decimal places. For example: 96 • 43 - 24 • 61 Another example: 96 • 43 - 24 • 61 5 - 1 96.43 24.61 71.82 Always start with the smallest units: hundredths or pennies: 3-1= 2 Move onto tenths or 10 pences: 40-60= Without going into negative numbers I have not got enough so I need to take 1 unit or £1 from my number. Now 1.40-0.60 = 0.80 Move onto units or £1: 5-4= 1 Move onto tens or £10: 90-20= 70 Towngate Primary Academy Times tables expectations Towngate Primary Academy believe that when children are proficient in times tables it enables them to calculate more efficiently. Expectations within each year group. Year 1 Children to be able to count in 2 and 10’s. Extension in 5’s. Year 2 Times tables 2, 5, 10. Summer term 3 and 4 and inverse calculations. Year 3 Times tables 6 ,7, 8 and 9 and inverse calculations. Year 4 Times tables – confident in all times tables and inverse calculations. Year 5 & 6 Rapid recall of all times tables and inverse. MULTIPLICATION Multiplication is taught as repeated addition10. Initially, children record their calculations using pictures. eg: How many legs on 5 horses? + 4 4 + + 4 + + 4 + + 4 + This can be modelled on the number line: 0 4 8 12 16 20 Before using the multiplication symbol 'x' children use the terms: 'groups of' 'lots of' 'sets of' 'bags of' etc emphasising 'of' eg 4 groups of 3, 4 lots of 3, 4 sets of 3 etc 4 groups of 3 make 12 altogether OR 3 + 3 + 3 + 3 = 12 Children are taught that when multiplying, the answer is the same whichever way round the multiplication is written: 10 repeated addition is the adding of two or more of the same number eg 3 x 5 is the same as 5 + 5 + 5 For example: 2 groups of 5 is the same as 5 groups of 2 2 x 5 = 5 x 2 = 10 This commutative11 property of multiplication is also taught through the use of 'arrays12' eg * * * * * * * * * * * * * * * * * * * * * * * * 3 rows of 4 = 4 rows of 3 3x4 = 4x3 = 12 The next stage is for children to use a number line for multiplication calculations: 1 group of 4 1 group of 4 1 group of 4 3 groups of 4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 group of 3 1 group of 3 1 group of 3 1 group of 3 4 groups of 3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 So 3 x 4 =12 and 4 x 3 = 12 When multiplying a 2-digit number by a 1-digit number children are taught to partition the 2-digit number (using arrow cards when necessary) and use a 'grid'. 11 see note on page 4 12 an array is a 2 dimensional picture of a multiplication or division fact eg 2 x 3 = 6 and 6 ÷ 2 =3 For example: 28 x 9 x 20 8 9 180 72 252 Children multiply the numbers, writing the answers in the correct parts of the grid. They find the answer by adding 180 and 72 N.B. Children should be encouraged to use column grid for addition. 100 80 + 70 2 100 + 150 + 2 = 252 The grid method is also used when multiplying a 3-digit number by a 1-digit number. 586 x 6 For example: x 500 80 6 6 3000 480 36 3000 400 80 + 30 6 3000 + 400 + 110 + 6 = 3516 3516 and when multiplying a pair of 2-digit numbers 47 x 23 For example: x 40 7 20 800 140 940 3 120 21 141 1081 800 +100 20 +100 40 + 20 1 1000 + 80 + 1 = 1081 Formal Method Multiplication To be taught when children are confident with simple grid calculations. For example: 55 x 24 = 55 x 24 20 4 x 5 = 20 (unit x unit) 200 4 x 50 = 200 (unit x tens) 100 20 x 5 = 100 (ten x unit) 1000 20 x 50 = 1000 (ten x ten) 1320 add columns already in place Children should begin by writing down x calculations they have done so mistakes can be easily identified. When accuracy improves these can be left out. 312 x 42 = For example: 312 x 42 4 20 600 80 400 12000 2x2=4 2 x 10 = 20 2 x 300 = 600 40 x 2 = 80 40 x 10 = 400 40 x 300 = 12000 1 1 13104 Multiplication of numbers with decimals Formal method of multiplying numbers with one decimal place This is introduced in the context of money to help with understanding. £22.3 x 4 For example: In the context of money: 22 is £22 and •3 is 30p 22.30 x 4 0 4 x 0 pennies = 0 1.20 4 x 0.30 = 1.20 (4 x 30p = £1.20) 8.00 4 x 2.00 = 8.00 80.00 89.20 4 x 20.00 = 80.00 Multiplication of numbers with 2 decimal places Again this is introduced in the context of money. £22.36 x 4 For example: In the context of money: 22 is £22 and 36 is 36p 22.36 x 4 24 4 x 0.06 (or 4 x 6p) = 0.24 1.20 4 x 0.30 (or 4 x 30p) = 1.20 8.00 4 x 2.00 = 8.00 80.00 89.20 4 x 20.00 = 80.00 DIVISION Division is the inverse13 of multiplication Children are introduced to the two aspects of division, namely: division as sharing division as making equal groups The appropriate approach will depend on the context of the problem presented to the children. At first, children use pictures to record their calculations: eg 'Share 12 buns among 4 people' 12 buns shared among 4 people = 3 buns each This is division by sharing. The answer is always the number of items in a group. eg How many groups of 4 cars can be made from 12 cars 13 Like addition and subtraction, division and multiplication are inverses (they undo each other's actions 12 cars can be made into 3 groups of 4 cars. This is division by grouping. The answer is always the number of groups. Children are taught to use a number line for division calculations. First they use a numbered line: eg There are 28 children. 4 children can fit into a car. How many cars are needed? 1 car 1 car 0 1 2 3 4 5 28 29 30 1 car 1 car 1 car 1 car 1 car 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 The children add up the number of jumps to find the number of cars ie 7 cars are needed. Children then use a blank number line and write on the significant numbers 1 car 0 1 2 3 4 5 29 30 1 car 1 car 1 car 1 car 1 car 1 car 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 267 28 Formal method of dividing Short Division: For example 152 ÷ 4 = 4 152 How many 4’s in 1 = 0 (record above the 1 so move the 1 onto the next number. 03 8 4 15 32 How many 4’s in 15 = 3 x 4 = 12 with 3 remainder which I put with my next number. How many 4’s in 32 = 4 x 8 = 32 Long division by chunking For example 972 ÷ 36 = To find out how many 36’s are in 972 by using key facts. Starting by using multiplication facts they know. 27 36 972 - 720 20 x 1 1 252 - 180 72 72 0 5x 2 x(add these) 27 Where remainders occur, children express them as fractions, decimals or use rounding depending upon the problem. For example 254 ÷ 17 = 16 14 r16 or 0.94 or 17 or 0.94 17 x 4 = 48 1 17 254 - 170 10 x 17 x 5 = 85 17 x 10 = 170 1 - 84 68 16 4 Where remainders occur, children express them as fractions, decimals or use rounding depending upon the problem. The Use of Calculators The calculator is a powerful and efficient tool. It has a strong part to play in subjects such as geography, history and science, since it allows children of primary age to make use of real data - often numbers with several digits - that they have gathered in their research or experiments, perhaps to work out a percentage, or to compare totals or proportions. In our school the calculator's main role in mathematics lessons will not be as a calculating tool, since children are still developing the mental calculation skills and written methods that they will need throughout their lives. But it does offer a unique way of learning about numbers and the number system14, place value15, properties of numbers, and fractions and decimals. If children are to use the basic facilities of a calculator constructively and efficiently, they need to be taught the technical skills they will require: the order in which to use the keys; how to enter numbers such as sums of money, measurements or fractions; how to interpret the display; how to use the memory; etc. Children need to learn when it is, and when it is not, appropriate to use a calculator and their first-line strategy should involve mental calculations16 wherever possible. For example, children might be taught that they can 'beat the calculator' if they can recall number facts rapidly. They should also have sufficient understanding of the calculation in front of them to be able to decide which method to use - mental, pencil and paper, or calculator. When they do use a calculator they should be able to draw on well established skills of rounding numbers and calculating mentally to gain a sense of the approximate size of the answer and have strategies to check and repeat the calculation if they are not sure whether it is right For these reasons, we would not normally use the calculator as part of Key Stage 1 mathematics but instead would emphasise oral work and mental calculation. But by the end of Key Stage 2 our children ought to have the knowledge and competence to use a calculator to work out, say, (56 + 97) ÷ (133 - 85) and round the answer to one decimal place17. They should also recognise that an approximate answer is 150 ÷ 50, or 3, and use this to check their calculation. 14our number system is based on counting in units, tens, hundreds etc as well as tenths, hundredths, thousandths etc 15see note on page 18 16mental calculations are done in the head rather than on paper 17the number 2.39 rounded to one decimal place would become 2.4